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- The following topics will be covered in this lecture:
- A review of the basics of continuous distributions
- The uniform distribution
- The normal distribution

Unlike

**discrete random variables**,**continuous random variables**can take on an**uncountably infinite number of possible values**.- This is to say that if \( X \) is a
**continuous random variable**, there is**no possible index set \( \mathcal{I}\subset \mathbb{Z} \)**which can enumerate the possible values \( X \) can attain. - For
**discrete random variables**, we could perform this with a possibly**infinite index set**, \( \{x_j\}_{j=1}^\infty \) - This has to do with how the
**infinity of the continuum \( \mathbb{R} \)**is actually**larger than**the**infinity of the counting numbers, \( \aleph_0 \)**; - in the
**continuum**you can**arbitrarily sub-divide the units of measurement**.

- This is to say that if \( X \) is a
These

**random variables are characterized**by a**distribution function**and a**density function**.Let

, then the mapping \[ F_X:\mathbb{R} \rightarrow [0,1] \] defined by \( F_X (x) = P(X \leq x) \), is called the **cumulative distribution function (cdf)**of the rv \( X \).A mapping \( f_X: \mathbb{R} \rightarrow \mathbb{R}^+ \) is called the

**probability density function (pdf)**of an rv \( X \) if \( f_X(x) = \frac{\mathrm{d} F_X}{\mathrm{d}x} \) exists for all \( x\in \mathbb{R} \); andand the

**density is integrable**, i.e., \[ \int_{-\infty}^\infty f_X (x) \mathrm{d}x \]**exists and takes on the value one**.

**Q:**we defined, \[ \begin{align} f_X(x) = \frac{\mathrm{d} F_X}{\mathrm{d}x} & & \text{ and }& & \int_{a}^b \frac{\mathrm{d}f}{\mathrm{d}x} \mathrm{d}x = f(b) - f(a) \end{align} \] how can you use the**definition above**and the**fundamental theorem of calculus**to**find another form for the CDF**?**A:**Notice that \( \frac{\mathrm{d} F_X}{\mathrm{d}x} \) means that the**CDF**can be**written in terms of the anti-derivative of the density**.- If
**\( s \)**and**\( t \)**are**arbitrary values**, the**definite integral**is written as

\[ \begin{align} \int_{s}^t f_X(x) \mathrm{d}x &= \int_{s}^t \frac{\mathrm{d} F_X}{\mathrm{d}x} \mathrm{d}x\\ &= F_X(t) - F_X(s) \\ & = P(X \leq t) - P(X \leq s) = P(s < X \leq t) \end{align} \]

- If we take a limit as \( s \rightarrow \infty \) we thus recover that

\[ \begin{align} \lim_{s\rightarrow - \infty} \int_{s}^t f_X(x) \mathrm{d} x & = \lim_{s \rightarrow -\infty} P(s < X \leq t) \\ & = P(X\leq t) = F_X(t) \end{align} \]

- Last week we discussed how the elementary properties of the
**probability distribution**of a**discrete rv**can be described by an**expectation**and a**variance**. - With respect to this, the only difference with
**continuous rvs**is in the use of**integrals**, rather than**sums**, over the possible values of the rv. - Let \( X \) be a
**continuous rv**with a**density function \( f_X(x) \)**– then the**expectation of \( X \)**is defined as \[ \mathbb{E}\left[X\right] = \int_{-\infty}^{+\infty} xf_X(x)\mathrm{d}x = \mu_X \] where \( f_X \) is the density function described before. - Note that the same interpretation of the expected value from discrete rvs applies here:
- We see
**\( \mathbb{E}\left[X\right]=\mu_X \)**as representing the**“center of mass” for the “density” curve \( f_X \)**. - We see
**\( \mathbb{E}\left[X\right]=\mu_X \)**as representing the**mean**that we would obtain if we could take**infinitely many independently replicated measurements of \( X \)**, and took the average of these measurements over all possible scenarios. **If the expectation of \( X \) exists**, the**variance**is defined as \[ \begin{align} \mathrm{var} \left(X\right)& = \mathbb{E}\left[\left(X − \mu_X \right)^2\right] \\ &=\int_{-\infty}^\infty \left(x - \mu_X\right)^2 f_X(x)\mathrm{d}x = \sigma_X^2 \end{align} \]

- Once again, this is a
**measure of dispersion**by**averaging the deviation**of each case from the mean in the square sense, weighted by the probability density.

- While the
**variance**is a**more “fundamental” theoretical quantity**for various reasons, in practice**we are usually concerned with**the**standard deviation**of the random variable \( X \), \[ \mathrm{std}(X)=\sqrt{\mathrm{var}\left(X\right)} = \sigma_X. \] - This is due to the fact that the
**variance \( \sigma^2_X \)**has the**units of \( X^2 \)**by the definition of the product. - For example, if the
**units of \( X \) are \( \mathrm{cm} \)**, then**\( \sigma_X^2 \) will be in \( \mathrm{cm}^2 \)**. - Taking a square root on the variance gives us the
**standard deviation \( \sigma_X \)**in the**units of \( X \) itself**.

While together the

**mean \( \mu_X \)**and the**standard deviation \( \sigma_X \)**give a picture of the**center**and**dispersion**of a probability distribution, we can analyze this in a different way.For example, while the mean is the notion of the “center of mass”, we may also be interested in

**where the upper and lower \( 50\% \) of values are separated**as a different notion of**“center”**.- The value that separates this upper and lower half
**does not need to equal the center of mass in general**, and it is known commonly as the**median**.

- The value that separates this upper and lower half
More generally, for any

**univariate cumulative distribution function \( F \)**, and for**\( 0 < p < 1 \)**, we can identify**\( p \) as a percent of the data that lies under the graph of a density curve**.- We might be interested in where the
**lower \( p \) area**is**separated from**the**upper \( 1-p \) area**.

- We might be interested in where the
The quantity \[ \begin{align} F^{-1}(p)=\inf \left\{x \vert F(x) \geq p \right\} \end{align} \] is called the

**theoretical \( p \)-th quantile or percentile of \( F \)**.The “\( \inf \)” in the above refers to the smallest possible quantity in the set on the right-hand-side.

We will usually refer to the

**\( p \)-th quantile as \( \xi_p \)**.\( F^{-1} \) is called the

**quantile function**.- Particularly, \( \xi_{-\frac{1}{2}} \) is known as the
**theoretical median**of a distribution.

- Particularly, \( \xi_{-\frac{1}{2}} \) is known as the